Nucleon transfer in reactions 3 He + 194 Pt, 45 Sc within time-dependent approach

Theoretical description of the experimental data on the formation of various isotopes in reactions ( 3 He + 194 Pt) and ( 3 He + 45 Sc) requires taking into account neutron and proton transfer channels, as well as fusion-evaporation channels. To calculate the probabilities of nucleon transfer as well as transfer cross sections the time-dependent Schrodinger equation (TDSE) has been solved numerically. Fusion-evaporation was taken into account using the statistical model code of the NRV web knowledge base. Results of calculations are in agreement with experimental data.


Introduction
The processes of neutron and proton transfer have been extensively studied both experimentally and theoretically. The theoretical description of experimental data on formation of gold isotopes in the ( 3 He+ 194 Pt)-reaction [1] and the 45 Ti isotope in the ( 3 He+ 45 Sc)-reaction [2] requires taking into account proton transfer channels. Considering these channels in the DWBA approximation provides (for the appropriate choice of parameters) satisfactory description of the experimental angular distributions for energies much higher than the Coulomb barrier [3]. For near-barrier energies, however, oscillations in angular distributions are weakly pronounced, complicating determination of the optical potential parameters. It is easier in this case to compare theoretical calculations and experimental data on the formation cross sections of the reaction products.
The cross section for the 194 Au isotope formation in the ( 3 He+ 194 Pt)-reaction was measured by means of activation in [1]. Possible mechanisms of the 194 Au isotope formation include proton capture by the target nucleus with the loss of a neutron due to its evaporation (after the excitation associated with proton capture), or to its capture by the projectile nucleus (with the transformation of 3 He into 3 H). Another possible mechanism of 194 Au isotope formation, fusion of nuclei with evaporation of two neutrons and a proton, is of low probability because of the high Coulomb barrier for protons. For the ( 3 He+ 45 Sc)-reaction [2], the probability of 45 Ti isotope formation in the fusion-evaporation channel is much higher because of the lower Coulomb barrier.
In this work, fusion-evaporation was considered using the numerical code of the NRV web knowledge base [4,5]. The numerical solution of the time-dependent Schrodinger equation taking into account spin-orbit interaction [6] was successfully used to calculate the cross sections of neutron transfer [7] and proton transfer [8] in reactions with the participation of 3 He nuclei. In this work, the time-dependent quantum approach is used to describe proton stripping from the 3 He nuclei in collisions with the 194 Pt and 45 Sc nuclei combined with neutron pickup from the target nuclei.

Theory
In [6][7][8], the time-dependent Schrodinger equation for protons or neutrons in combination with the classical equations of motion for nuclear centers of mass was used to describe transfer channels in collisions of atomic nuclei. Here, I r 1 (t) , I r 2 (t) are the centers of nuclei with masses m 1 , m 2 and V 12 (r) is the potential energy of interaction of these nuclei, m is a proton (neutron) mass, ψ is a spinor wave function. Before touching of the surfaces of nuclei, the potential energy The initial condition for the proton wave function of the 3 He projectile nucleus was defined taking into account the long-range character of the Coulomb interaction with the target nucleus. The proton wave function in an isolated projectile nucleus at a finite distance from the target nucleus was preliminarily subjected to slow (adiabatic) switching of the Coulomb interaction with the target nucleus. Thus, the effects of the proton cloud polarization were taken into account in the initial condition.

Results for ( 3 He+ 194 Pt) reaction
The initial condition for the proton wave function of the 3 He projectile nucleus was based on the shell model with the mean field potential V 1 (r) of the 3 He nucleus given in [7]. It differs from the Woods-Saxon potential by the maximum at zero providing better agreement with the experimental charge distribution [4]. The proton and neutron energy levels ε of the 3 He nucleus are shown in Figures 1(a) and 1(d), respectively. The Woods-Saxon form V 2 (r) was used as the nuclear part of the potential of proton (neutron) interaction with the heavy target nucleus. The values of the parameters were chosen from the condition of equality of the theoretical and experimental values of the energy E 8 of proton or neutron separation from the target [8].
After proton transfer to the 194 Pt nucleus, the 195 79 Au 116 nucleus is formed. The upper single-particle proton levels of the 195 Au nucleus are shown in Figure  1(b). The neutron levels of the 194 Pt nucleus near the uppermost occupied level (Fermi level) are shown in Figure 1(e).
The typical pattern of the change in the proton probability density upon its stripping in the 3 He+ 194 Pt collision is shown in Figure 2. Here and below, the effective radius of the 3 He nucleus is − R 1 =2.2 fm, which is somewhat larger than its root-mean-square charge radius R 1ch ≈ 2 fm [4], and the effective radius of the 194 Pt nucleus was determined according to the usual formula For near-barrier energies [ Figure 2a and Figure 2c], the structure of the probability density in the target nucleus corresponds to the dominant stripping of the neutron to the state with close energy 3 s1/2 [see Figure 1(b)] of the produced 195 Au nucleus. Under the conditions of the slow relative motion of the colliding nuclei and the fast motion of the transferred proton, we observe an adiabatic regime [9] of probability density redistribution with the population of two-center states. In this case, only the high-lying states 2f 7/2 and 1h 9/2 can be populated, along with the state 3 s1/2 . The population of such states is higher in the grazing collisions at above-barrier energies [Figures 2(d)-2(f)], as can be seen from the increased probability density extending beyond the boundaries of the nucleus. As nuclei fly away and the distance between them grows, excitation can be reduced by the emission of both protons and neutrons. This gives the first mechanism of 194 Au isotope formation. The probability can be estimated using the formula where C ≤ 1 is a variable (adjustable) parameter, p 0 is the probability for the proton from the 3 He nucleus to enter the region of the 194 Pt nucleus (not farther than the approximate radius of nuclear forces R NN ≈ 3 fm from its surface) in the time the projectile nucleus passes the target nucleus; and p (−) is the sum of populations of the initially free levels with energy ε <0 in the 195 Au nucleus: where a k are the coefficients of expansion of the time-dependent proton wave function ( Figure 2) into the wave functions of the unoccupied single-particle states of the target nucleus. The difference p 0 − p (−) is the probability of occupation of the quasistationary states with energy ε >0 in the potential well of the target nucleus surrounded by the Coulomb barrier with the lifetimes exceeding the interaction time of the nuclei. The residual interaction between the proton in these states and a neutron of the target nucleus may lead to the transition of this proton to unoccupied levels of the target nucleus accompanied by energy transfer to the neutron exceeding its separation energy. The second mechanism of 194 Au nucleus formation is a combination of neutron transfer from the 194 Pt nucleus to the 3 He nucleus and proton transfer in the opposite direction. The probability of this process is the product of the probability p (−) for proton stripping to the unoccupied single-particle states of the target nucleus and the probability p ′ 0 for the neutron to enter the region of the projectile nucleus [7] During the transformation of the 3 He nucleus into 3 H, the mean field of the shell model can be assumed to vary slightly. The close energies of the neutron level in the projectile nucleus and the upper neutron levels of the 194 Pt nucleus [see Figure 1d and Figure 1e] in this case increases the probability of transfer between them. The most probable is neutron pickup from level 3 p 3/2 due to lower centrifugal barrier. Although the probability of neutron pickup from the broad potential well of the target nucleus to the narrow well of the projectile nucleus is low, the large number of neutrons (16) on the levels near the Fermi level increases the probability of pickup of one of them, and thus makes the contribution of the second mechanism of 194 Au formation noticeable.
The probabilities of the above processes are shown in Figure 3a as functions of minimum distance R min (b, E) between the centers of the colliding nuclei, where b is the collision impact parameter, and E is the center-of-mass energy. The probability logarithms can be smoothed by linear dependence with parameters A and B depending weakly on energy. The cross section of proton transfer (or a combination of proton and neutron transfer) was calculated by integrating the corresponding probability over the impact parameters of grazing collisions b > b min : The results of calculating the cross section of 194 Au isotope formation via proton stripping with neutron evaporation from the target nucleus and via proton stripping with neutron pickup are shown in Figure 3b along with the negligible contribution of the fusion-evaporation mechanism.

Results for ( 3 He+ 45 Sc) reaction
The nucleus 45 21 Sc 24 has an unpaired proton on shell 1 f 7/2 . The experimental angular distributions for inelastic scattering given in [3] show that after proton transfer, the ground state of the 46 22 Ti 24 nucleus with zero spin and two paired outer protons (on shell 1 f 7/2 ) is produced with rather low probability. Therefore, the Ti nucleus is formed in exited states, 46 22 Ti * 24 , with unpaired outer protons. In calculations, we used upper single-particle proton levels of the 46 22 Ti * 24 nucleus which were similar to the levels of the 45 21 Sc 24 nucleus; they are shown in Figure  1(c). A typical pattern of the change in the proton probability density upon its stripping in the collision ( 3 He+ 45 Sc) nuclei is shown in Figure 4.
An adiabatic scenario [9] of probability density redistribution with population of two-center states is observed for near-barrier energies [ Figure 4a and Figure 4c]. We can see that the structure of the probability density in the target nucleus corresponds to the dominant proton stripping to states 2 p 3/2 , 2 p 1/2 of the formed 46 Ti nucleus because the angular dependence of the probability density corresponds to the Legendre polynomial P 1 (cosθ) =cos θ and the radial dependence has one extra node in addition to the node at the center of the target nucleus (at r=0). Similar behavior was observed in [7] for neutron stripping from the state 1 s1/2 of the 3 He nucleus to the state 2 p 3/2 of the Sc nucleus. Since the energies of these levels [see Figure 1d and Figure 1f] are close, the probability of neutron transfer is high. The probability of proton transfer is much lower because of the larger difference between the energies of the initial and final levels [see Figures 1a and Figure 1c]. The probability of neutron pickup by the 3 He nucleus from the lower levels of the 45 Sc nucleus in combination with proton stripping is extremely low for the same reason. In the case of grazing collisions with above-barrier energies [ Figure 4d and Figure 4f], the rate of proton probability density redistribution is comparable to the relative velocity of the nuclei. This results in a nonadiabatic proton transfer regime [9] with the formation of highly excited states. As nuclei go away from each other, excitation can be reduced by the emission of both protons and neutrons. The probabilities of these processes are shown in Figure 5a as functions of the minimum distance between the centers of the colliding nuclei. The results of calculating the cross section of 45 Ti isotope formation via proton stripping with neutron evaporation from the target nucleus and via proton stripping with neutron pickup are shown in Figure 5b along with the contribution from the fusion-evaporation mechanism which is the main mechanism in this case.

Discussion
Study of proton and neutron transfer can provide information on the properties of nuclear states of predominantly single-particle nature. For instance, the sufficiently large values of the transfer cross section at near-barrier energies indicate that the initial and final energy levels of the transferred nucleon are close (e.g., for the 3 He+ 194 Pt reaction). Vice versa, the low values of the transfer cross section at near-barrier energies indicate large difference between the initial and final energy levels of the transferred nucleon (e.g. for the 3 He+ 45 Sc reaction).

Conclusions
The experimental difference in the near-barrier energy dependence of formation cross sections for the 194 Au isotope in the ( 3 He+ 194 Pt) reaction and the 45 Ti isotope in the ( 3 He+ 45 Sc) reaction was explained by the difference in the proton and neutron shells of the target nuclei determining different character of evolution of the probability density for the proton of the projectile nucleus and the neutron of the target nucleus during the collision. Experimental data on the formation cross sections for a number of isotopes can be used for validation and refinement of theoretical models of nucleon transfer, nuclear fusion, and pre-equilibrium processes. The time-dependent approach used in this work can also be applied to calculation of the proton transfer cross sections in reactions with nuclei having proton halos (e.g. 8 B) and the cross sections of charged cluster (deuterons, α -clusters) transfer.